1. Field of Invention
The present invention relates to modeling behavior of distributed time invariant systems, and is more particularly related to the modeling of distributed time invariant devices used in the design of integrated circuits. The present invention is yet further related to the preparation of reduced order models of distributed time invariant devices and their use in simulators.
2. Discussion of Background
Design of modern high-performance electronic systems such as RF circuits, optical transceiver ICs, and global digital signal interconnect requires careful attention to physical modeling so that the intrinsic physical limitations of implementation processes can be accounted for, and undesirable effects such as signal integrity problems and excessive electromagnetic interference (EMI) can be minimized. Effective design of complicated systems requires simple models, and this simplicity requirement conflicts with the need for accuracy, as a high degree of physical fidelity can only be achieved by detailed analysis, such as with electromagnetic field solvers. Model reduction is now a standard procedure for obtaining simple models of complicated physical systems.
The design process of modern systems can be broken down into 3 levels of abstraction. First, a system designer who is responsible for an overall design (e.g., cell phone, wireless LAN, a chipset, etc.). The system designers are working at the very highest level of abstraction, and typically think in terms of software, what data needs to be transmitted between various components of a design, and processing of the data. At a second, lower level of abstraction, which we will call a circuit design level, we find engineers that are designing the circuits that actually do the operations envisioned by the system designers. At the circuit design level, the engineers are thinking in terms of things like transistors, resistors, inductors, and other components. The circuit design engineers run the simulation tools like SPICE, or other SPICE class simulations, signal processing performed in C language, etc., these tools are often based on solving differential equations or ordinary differential equations.
And then, at a lowest level of abstraction, which can be referred to as the physical level, scientists and physicists analyze and describe an actual physical object (e.g., properties of a piece of metal, or silicon). The physicists task is to describe the piece of metal, for example by calling it a piece of metal, noting its size shape, etc, and writing down Maxwell's equations for that piece of metal and describe it in detail. Here, the physicist is describing the object by calling it a component name, such as a resistor or inductor, and making it an abstract object that can be used by the engineers at the next higher design level.
The description of the metal then provides a very detailed account of the properties of the component it embodies. The properties are typically described in a set of ordinary and partial differential equations based on the geometry and properties of the metal. The set of equations is large and complicated and are needed because a very accurate description of that particular component is critical to the functioning of a system being designed at the higher levels. Further, the component may need to be placed in some structure that is, for example, a package for an IC or some very complicated structure like an integrated inductor on a chip.
Now, turning back to the highest level of abstraction, the system designers certainly have some idea that maybe there is an inductor or other components in the system, but they would not likely be involved about the properties or placement of those components. Then, at the circuit level, the engineers know about the placement and certain qualities of the inductors, and are aware that the inductor is a physical component. And, so the entire process can be viewed as starting from the physical level, a design begins from a very detailed description of the object and then to generate some model that we call a macromodel, or reduced order model, (either by hand or via an automated process such as model reduction) so that the circuit level engineers can generate circuit designs which are used by the system designers to complete a system.
So, starting with the physical geometry of the metal and the properties of the metal, its shape, etc., the physicist is figuring out how it is going to react when placed in either a resistor, inductor, or other component configuration. The problem is that the description that accurately describes these properties might be 10K equations to describe one component. And, in a particular circuit there may be 10K RLCs that are implemented. The resultant 10,000, 100,000, or one million different equations need to be evaluated, which is a cumbersome and not really practical even with modern high speed processing capabilities.
The physical level differential equation descriptions are very accurate, and what is needed is to get from that very accurate but cumbersome description to a simpler more manageable formula (converting a very detailed level geometrical to a more abstract circuit type description) that can be simulated using currently available technologies. There are many ways to generate such simplified models, the most common procedure being for a skilled engineer to craft each model by hand. Because by hand model generation is very labor intensive and requires highly skilled and experienced individuals to perform the task, it is attractive to seek an automated procedure for generating the macromodels. An automatic procedure that generates reduced order models from detailed physical equations is called a model order reduction procedure.
Thus, model reduction forms a bridge between the detailed physical level of analysis and the circuit or system level where design is performed by extracting a simple behavioral model from the complicated lower level description. Model reduction addresses three primary issues, each of which can be identified with a stage of evolution of the research field.
First, the models that are produced must be accurate. The attractiveness of model reduction procedures, over other behavioral modeling approaches, is that they offer a systematic way to control and predict the accuracy of the reduced models. This requirement led to the proliferation of moment-matching approaches.
Second, it must be possible to generate a model, of arbitrary order, in a numerically stable way. Early moment-matching approaches suffered from numerical instability problems that prevented reliable computation of high-order models. This requirement eventually led to the Krylov-subspace algorithms such as PVL that stabilize the interpolation procedures.
Third, the models that are generated must be well-behaved when embedded into a simulation tool with models of other physical elements. While not possible to guarantee that the reduced model will not have unintended consequences in every possible simulation, a reasonable procedure is to require that the models themselves do not possess non-physical properties. For example, components such as interconnect do not generate energy; they are passive. Thus, much work in model reduction in the past several years has been concerned with passivity-preserving procedures such as those based on congruence transforms.
In the past, model order reduction was performed in a number of ways. For example, engineers have simulated the descriptions and tried to perform some type of curve fitting (a popular method), which basically comprises plotting out the model as a function of frequency and then trying to fit a curve through it. Or, in another method plotting a transfer function and maybe some of its derivative around a point of interest. Each of these methods try to approximate the transfer function with a smaller set of equations or a function. However, these standard curve fitting methods are used only in limited or restricted contexts and have some large problems associated with them. For example, when curve fitting is performed around a point of interest or for certain frequencies, the results can be quite good. But when placed in a simulation tool, and particularly at different frequencies, the behavior of the curve fit equations are not entirely predictable nor dot they necessarily behave like the materials which they are to be approximating. Changes in behavior can be so different that in some scenarios passive materials can have curve fit models that appear active. As another example, rational function approximation of data from functions exhibiting sharp resonant behavior (i.e. near where the denominator of the rational function approaches zero) is problematic because of the potentially high data density needed near the resonance, as well as the increased dynamic range of the data.
Referring now to FIG. 1, there is provided an illustration of different types of equations and different algorithms 100 that have been utilized to attempt to solve the order reduction problems. FIG. 1 illustrates the different types of systems which are linear, non-linear, distributed, time varying, and time invariant. The systems themselves can generally be divided into lumped and distributed systems. Broadly speaking, the type of models needed to represent a system of interest is dependent on whether the system of interest is lumped or distributed.
In one previous system, described in Phillips, U.S. Pat. No. 6,349,272, entitled “Method and system for modeling time-varying systems and non-linear systems,” the contents of which are incorporated herein by reference, there is described a reduction method that is directed towards a special class of systems called time varying systems 110.
Other systems that address model reduction, but for a limited class of designs or properties, include Cullum et al. (U.S. Pat. No. 6,188,974) (e.g., 120), Feldmann et al. (U.S. Pat. No. 5,689,685) (e.g., 130), Ngyuen et al. (U.S. Pat. No. 5,920,484), and Celik et al. (IEEE Trans. CAD, 16(5). pp. 485-496). However, each of these systems are limited in that they only operate on limited classes of systems, or are otherwise limited to low frequency analysis.
For RLC circuits many methods exist that can generate adequate reduced model representations. Likewise, it is known how to formulate RLC circuit equations such as to guarantee stability and positive-realness of the reduced models, which is sufficient to guarantee that the reduced models belong to the same class (passive systems) as the full models from which they were derived. However, not all systems of interest in IC design are lumped. At high frequencies, distributed effects become necessary to model. For example, full-wave integral-equation based field solvers generate frequency-dependent matrices that cannot be used as inputs to the standard model-reduction procedures that require constant matrix representations. In fact, most integral formulations that model either frequency or spatial variation (such as those with layered-media Green functions and most surface integral formulations) contain distributed parameter descriptions. However, very few methods exist for reduction of distributed systems, and are either limited to very specific geometric structures (e.g., Celik et al. IEEE Trans. CAD, 16(5). pp. 485-496) or have no guarantees as to passivity or positive-realness properties (Cullum et al. U.S. Pat. No. 6,188,974).